Accounting and the Time Value of Money Chapter 7 Accounting and the Time Value of Money ACCT-3030

1. Basics Study of the relationship between time and money Money in the future is not worth the same as it is today because if you had the money today you could invest it and earn interest not because of risk or inflation ACCT-3030

1. Basics Based on compound interest not simple interest ACCT-3030

1. Basics Examples of where TVM used in accounting Notes Receivable & Payable Leases Pensions and Other Postretirement Benefits Long-Term Assets Shared-Based Compensation Business Combinations Disclosures Environmental Liabilities Revenue Recognition ACCT-3030

1I. Future Value of Single Sum The amount a sum of money will grow to in the future assuming compound interest Can be compute by formula: tables: calculator: TVM keys FV = PV ( 1 + i )n FV = PV x FVIF(n,i) (Table 7A.1) FV = future value n = periods PV = present value i = interest rate FVIF = future value interest factor ACCT-3030

1I. Future Value of Single Sum Example If you deposit $1,000 today at 5% interest compounded annually, what is the balance after 3 years? ACCT-3030

1I. Future Value of Single Sum Calculate by hand Event Amount Deposit 1-1-x1 $ 1,000.00 Year 1 interest (1000 x .05) 50.00 End of Year 1 Amount 1,050.00 Year 2 interest (1050 x .05) 52.50 End of Year 2 Amount 1102.50 Year 3 interest (1102.50 x .05) 55.13 End of Year 3 Amount 1157.63 ACCT-3030

1I. Future Value of Single Sum Calculate by formula FV = 1,000 (1 + . 05)3 = 1,000 x 1.15763 = 1,157.63 ACCT-3030

1I. Future Value of Single Sum Calculate by table FV = 1,000 x Table factor for FVIF(3, .05) = 1,000 x 1.15763 = 1,157.63 ACCT-3030

1I. Future Value of Single Sum Calculate by calculator Clear calculator: 2nd RESET; ENTER; CE|C and/or: 2nd CLR TVM 3 N 5 I/Y 1,000 +/- PV CPT FV = 1,157.63 ACCT-3030

1I. Future Value of Single Sum Additional example If you deposit $2,500 at 12% interest compounded quarterly, what is the balance after 5 years? less than annual compounding so adjust n and i n = 20 periods i = 3% 2,500 x 1.80611 = 4,515.28 20N; 3 I/Y; -2500 PV; CPT FV = 4,515.28 ACCT-3030

1II. Present Value of Single Sum Value now of a given amount to be paid or received in the future, assuming compound interest Can be compute by formula: tables: calculator: TVM keys PV = FV · 1/( 1 + i )n PV = FV x PVIF(n,i) (Table 7A.2) FV = future value n = periods PV = present value i = interest rate PVIF = present value interest factor ACCT-3030

1II. Present Value of Single Sum Example If you will receive $5,000 in 12 years and the discount rate is 8% compounded annually, what is it worth today? ACCT-3030

1II. Present Value of Single Sum Calculate by formula PV = 5,000 · 1/(1 + . 08)12 = 5,000 x .39711 = 1,985.57 ACCT-3030

1II. Present Value of Single Sum Calculate by table PV = 5,000 x Table factor for PVIF(12, .08) = 5,000 x .39711 = 1,985.57 ACCT-3030

1II. Present Value of Single Sum Calculate by calculator Clear calculator 12 N 8 I/Y 5,000 FV CPT PV = 1,985.57 ACCT-3030

1II. Present Value of Single Sum Additional example If you receive $1,157.63 in 3 years and the discount rate is 5%, what is it worth today? n = 3 periods i = 5% 1,157.63 x .863838 = 1,000.00 3 N; 5 I/Y; 1157.63 FV; CPT PV = -1,000.00 ACCT-3030

1V. Unknown n or i Example 1 If you believe receiving $2,000 today or $2,676 in 5 years are equal, what is the interest rate with annual compounding? PV = FV x PVIF(n, i) 2,000 = 2,676 x PVIF(5, i) PVIF(5, i) = 2,000/2,676 = .747384 find above factor in Table 2: i ≈ 6% 5 N; -2,000 PV; 2,676 FV; CPT 1/Y = 6.00% ACCT-3030

1V. Unknown n or i Example 2 Same as last problem but assume 10% interest with annual compounding is the appropriate rate and calculate n. PV = FV x PVIF(n, i) 2,000 = 2,676 x PVIF(n, 10%) PVIF(n, 10%) = 2,000/2,676 = .747384 find above factor in Table 2: n ≈ 3 years 10 I/Y; -2,000 PV; 2,676 FV; CPT N = 3.06 years ACCT-3030

V. Annuities Basics annuity ordinary annuity annuity due a series of equal payments that occur at equal intervals ordinary annuity payments occur at the end of the period annuity due payments occur at the beginning of the period ACCT-3030

V. Annuities Ordinary annuity – payments at end Present Value |_____|_____|_____|_____|_____| Year 1 Year 2 Year 3 Year 4 Year 5 Pmt 1 Pmt 2 Pmt 3 Pmt 4 Evaluate PV ACCT-3030

V. Annuities Annuity due – payments at beginning Present value |_____|_____|_____|_____|_____| Year 1 Year 2 Year 3 Year 4 Year 5 Pmt 1 Pmt 2 Pmt 3 Pmt 4 Evaluate PV ACCT-3030

V. Annuities For Future Value of an annuity more difficult Determine whether the annuity is ordinary or due based on the last period if evaluate right after last pmt – ordinary if evaluate one period after last pmt – due An important part of annuity problems is determining the type of annuity ACCT-3030

V. Annuities Ordinary annuity – payments at end Future Value |_____|_____|_____|_____|_____| Year 1 Year 2 Year 3 Year 4 Year 5 Pmt 1 Pmt 2 Pmt 3 Pmt 4 Evaluate FV ACCT-3030

V. Annuities Annuity due – payments at beginning Future Value (evaluate 1 period after last payment) |_____|_____|_____|_____|_____| Year 1 Year 2 Year 3 Year 4 Year 5 Pmt 1 Pmt 2 Pmt 3 Pmt 4 Evaluate FV ACCT-3030

V. Annuities Tables available in book for Future Value of Ordinary Annuity (Table 7A.3) Future Value of Annuity Due (Table 7A.4) Present Value of Ordinary Annuity (Table 7A.5) Present Value of Annuity Due (Table 7A.6) Sometimes not all tables are provided and you must use what is given and make the appropriate adjustment. ACCT-3030

V. Annuities Annuity table factors conversion Use calculator to calculate FV of annuity due look up factor for FV of ordinary annuity for 1 more period and subtract 1.0000 to calculate PV of annuity due (can use table) look up factor for PV of ordinary annuity for 1 less period and add 1.0000 Use calculator change calculator to annuity due mode 2nd BEG; 2nd SET; 2nd QUIT to change back to ordinary annuity mode 2nd BEG; 2nd CLR WORK; 2nd QUIT (or 2nd RESET) ACCT-3030

V1. Future Value of Annuity Can be calculated by formula: table: calculator: TVM keys (1 + i)n - 1 FVA(ord) = Pmt ----------------- i FVA(ord or due) = Pmt x FVIFA(ord or due) (n, i) FV = future value n = periods PV = present value i = interest rate FVIF = future value interest factor ACCT-3030

V1. Future Value of Annuity Can be calculated by formula: (1 + i)n - 1 FVA(due) = Pmt --------------- x (1 + i) i FV = future value n = periods PV = present value i = interest rate FVIF = future value interest factor ACCT-3030

V1. Future Value of Annuity Example Find the FV of a 4 payment, $10,000, ordinary annuity at 10% compounded annually. (You could treat this as 4 FV of single sum problems and would get correct answer but that method is omitted.) ACCT-3030

V1. Future Value of Annuity Calculate by formula FVA-ord = 10,000 ----------- = 10,000 x 4.6410 = 46,410 (1 + .1)4 - 1 .1 ACCT-3030

V1. Future Value of Annuity Calculate by table (Table 6-3) FVA-ord = 10,000 x FVIFA-ord (4, .10) = 10,000 x 4.64100 = 46,410 ACCT-3030

V1. Future Value of Annuity Calculate by calculator 4 N; 10 I/Y; -10000 PMT; CPT FV 46,410 ACCT-3030

V1. Future Value of Annuity Additional examples Find the FV of a $3,000, 15 payment ordinary annuity at 15%. FVA-ord = 3,000 x FVIFA-ord (15, .15) = 3,000 x 47.58041 = 142,741 15 N; 15 I/Y; -3000 PMT; CPT FV = 142,741 ACCT-3030

V1. Future Value of Annuity Additional examples Find the FV of a $3,000, 15 payment annuity due at 15%. (table – look up 1 more period -1.0000) FVA-ord = 3,000 x FVIFA-due (15, .15) = 3,000 x 54.71747 = 164,152 2nd BGN; 2nd SET; 2nd QUIT 15 N; 15 I/Y; -3000 PMT; CPT FV = 164,152 ACCT-3030

VI1. Present Value of Annuity Can be calculated by formula: table: calculator: TVM keys 1 – (1/(1 + i)n) PVA(ord) = Pmt --------------------- i PVA(ord or due) = Pmt x PVIFA(ord or due) (n, i) FV = future value n = periods PV = present value i = interest rate PVIF = present value interest factor ACCT-3030

VI1. Present Value of Annuity Can be calculated by formula: 1 – (1/(1 + i)n) PVA(due) = Pmt --------------------- x (1 + i) i FV = future value n = periods PV = present value i = interest rate PVIF = present value interest factor ACCT-3030

VI1. Present Value of Annuity Example What is the PV of a $3,000, 15 year, ordinary annuity discounted at 10% compounded annually? ACCT-3030

VI1. Present Value of Annuity Calculate by formula PVA-ord = 3,000 ---------------- = 3,000 x 7.60608 = 22,818 1 – (1/(1 + .10)15 .10 ACCT-3030

VI1. Present Value of Annuity Calculate by table (Table 6-4) PVA-ord = 3,000 x PVIFA-ord (15, 10) = 3,000 x 7.60608 = 22,818 ACCT-3030

VI1. Present Value of Annuity Calculate by calculator 15 N; 10 I/Y; -3000 PMT; CPT PV 22,818 ACCT-3030

VI1. Present Value of Annuity Additional examples Find the PV of a $3,000, 15 payment annuity due discounted at 15%. PVA-due = 3,000 x PVIFA-due (15, .15) = 3,000 x 6.72488 = 20,175 2nd BGN; 2nd SET; 2nd QUIT 15 N; 15 I/Y; -3000 PMT; CPT PV = 20,173 ACCT-3030

VI1. Present Value of Annuity Additional examples If you were to be paid $1,800 every 6 months (at the end of the period) for 5 years, what is it worth today discounted at 12%? PVA-ord = 1,800 x PVIFA-ord (10, .06) = 1,800 x 7.36009 = 13,248 10 N; 6 I/Y; -1800 PMT; CPT PV = 13,248 ACCT-3030

VI1. Present Value of Annuity Additional examples If you consider receiving $12,300 today or $2,000 at the end of each year for 10 years equal, what is the interest rate? 12,300A-ord = 2,000 x PVIFA-ord (10, i) PVIFA-ord (10, i) = 12,300/2,000 = 6.15000 i ≈ 10% 10 N; -2000 PMT; PV = 12300; CPT I/Y = 9.98% ACCT-3030